She said running in the by-election was the "farthest thing from my mind" until Mr Turnbull was dumped as prime minister.

"Everywhere I went in the streets in the eastern suburbs, somebody would come up to me and say, 'Would you please have a run at Wentworth?'," Dr Phelps told the ABC today.

Dr Phelps says her first mission is to get refugee children off Nauru if she is confirmed to have won the seat.

Dr Phelps told Weekend Today she would stick to her campaign promises.

"The things I was talking about during the campaign are really important - the big issues of national importance," she said this morning.

"For example getting kids off Nauru. That's first order of business. I don't think the Australian people can tolerate that position any longer."

Despite heading towards a minority government, Mr Morrison has talked up the coalition's relationship with the crossbenchers.

"What I will continue to do is be working closely with the crossbenchers, as I have been doing, because ... we have been at 75 (seats), not 76, since the former prime minister resigned," Mr Morrison said.

Reaction to the by-election result from crossbench members has been mixed, with Bob Katter and Rebekha Sharkie joining Dr Phelps in saying they would prefer to see the government run its full term.

Independent Andrew Wilkie said he would not guarantee confidence, while fellow crossbencher Cathy McGowan is yet to comment.

But Greens MP Adam Bandt said an election had to be called and the "the sooner we turf out this rotten government, the better".

Dr Phelps said she would set two priorities as a MP: action on climate change and getting asylum seeker children off Nauru.

She might also lend support for a national integrity commission that has been promoted by Labor, the Greens and crossbenchers for some time.